526 Dr. J. R. Airey on Bessel and Neumann 



the one indicated at the end o£ this paper, the first roots of 

 J 10 (s) and J 1( )oo(» are 14-4755 and 1018'62. 



The value of J n (n-\- k) can be found directly from (13) 

 when k is larger than unity, if n is large. For example *, 

 J 18 (20) = 0* 25109. This result is more easily and accurately 

 obtained from J 19 (20) and J 20 (20). 



The Neumann Functions Gr n (z) and Y n (z). 

 The function j 



p n _ 7T \~ J _ n (z) — COS WIT J n (z) ~| 



" W ~2L sinrwr J' 



Substituting 



J -,(..)-J- 3 [© i ~.Kl-)-J„© i KD— (¥-)■■].(") 



and making n integral, we get 



«.<^[©MDn4©MD-^oGK) 



-™© Vr ©+-]- <"> 



The following asymptotic series represents G n (z) when 

 the argument z=n-\ k : — 



«-e>=lMIM»-M-!Ni)-H3MD 

 -SMK)* ■■■]. • ■ • w 



the coefficients B , B 1? &c, having the same meaning as 

 before. In the special case where n is the argument and 

 n — 1 the order of the function, 



^•> = i«) - ©Mi) - AM + a©md 



8100 W\3/ 113400 V»/ 1,3 / 



♦*&»#'(&■■] < 19 > 



i * Phil. Mag. July 1909, p. 17. 



' t Gray & Mathews, ' Bessel Functions/ p. 242. 



